CHAPTER 17 MAGNETIC DIPOLE MOMENT

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1 1 CHAPTER 17 MAGNETIC DIPOLE MOMENT 17.1 Intoduction A numbe of diffeent units fo expessing magnetic dipole moment (heeafte simply magnetic moment ) ae commonly seen in the liteatue, including, fo example, eg G 1, G cm, Oe cm, T m, A m 2, J T 1. It is not always obvious how to convet fom one to anothe, no is it obvious whethe all quantities descibed as magnetic moment efe to the same physical concept o ae dimensionally o numeically simila. It can be almost an impossibility, fo example, to wite down a list of the magnetic moments of the planets in ode of inceasing magnetic moment if one efes to the divese liteatue in which the moments of each of the nine planets ae quoted in diffeent units. This chapte exploes some of these aspects of magnetic moment. In pevious chaptes, I have used the symbols p e and p m fo electic and magnetic dipole moment. In this chapte I shall be concened exclusively with magnetic moment, and so I shall dispense with the subscipt m The SI Definition of Magnetic Moment If a magnet is placed in an extenal magnetic field B, it will expeience a toque. The magnitude of the toque depends on the oientation of the magnet with espect to the magnetic field. Thee ae two oppositely-diected oientations in which the magnet will expeience the geatest toque, and the magnitude of the magnetic moment is defined as the maximum toque expeienced by the magnet when placed in unit extenal magnetic field. The magnitude and diection of the toque is given by the equation The SI unit fo magnetic moment is clealy N m T 1. τ = p B If an electic cuent I flows in a plane coil of aea A (ecall that aea is a vecto quantity hence the boldface), the toque it will expeience in a magnetic field is given by This means that the magnetic moment of the coil is given by τ = I A B p = IA Thus the unit A m 2 is also a coect SI unit fo magnetic moment, though, unless the concept of cuent in a coil needs to be emphasized in a paticula context, it is pehaps bette to stick to N m T 1.

2 2 While J T 1 is also fomally dimensionally coect, it is pehaps bette to estict the unit joule to wok o enegy, and to use N m fo toque. Although these ae dimensionally simila, they ae conceptually athe diffeent. Fo this eason, the occasional pactice seen in atomic physics of expessing magnetic moments in MeV T 1 is not entiely appopiate, howeve convenient it may sometimes seem to be in a field in which masses and momenta ae often conveniently expessed in MeV/c 2 and MeV/c. It is clea that the unit T m, often seen fo magnetic moment is not dimensionally coect fo magnetic moment as defined above, so that, whateve quantity is being expessed by the often-seen T m, it is not the conventionally defined concept of magnetic moment. The magnetization M of a mateial is defined by the equation B = µ 0 (H + M) Equations and fo the definitions of magnetic moment and magnetization ae consistent with the altenative concept of magnetization as magnetic moment pe unit volume. 17. The Magnetic Field on the Equato of a Magnet By the equato of a magnet I mean a plane nomal to its magnetic moment vecto, passing though the mid-point of the magnet. The magnetic field at a point at a distance on the equato of a magnet may be expessed as a seies of tems of successively highe powes of 1/ (the fist tem in the seies being a tem in ), and the highe powes decease apidly with inceasing distance. At lage distances, the highe powes become negligible, so that, at a lage distance fom a small magnet, the magnitude of the magnetic field poduced by the magnet is given appoximately by B µ 0 p = π Fo example, if the suface magnetic field on the equato of a planet has been measued, and the magnetic popeties of the planet ae being modelled in tems of a small magnet at the cente of the planet, the dipole moment can be calculated by multiplying the suface equatoial magnetic field by µ 0 /(4π) times the cube of the adius of the planet. If B, µ 0 and ae expessed espectively in T, H m 1 and m, the magnetic moment will be in N m T 1.

3 17.4 CGS Magnetic Moment, and Lip Sevice to SI Equation is the equation (witten in the convention of quantity calculus, in which symbols stand fo physical quantities athe than fo thei numeical values in some paticula system of units) fo the magnetic field at a lage distance on the equato of a magnet. The equation is valid in any coheent system of units whateve, and its validity is not esticted to any paticula system of units. Example of systems of units in which equation ae valid include SI, CGS EMU, and Bitish Impeial Units. If CGS EMU ae used, the quantity µ 0 /(4π) has the numeical value 1. Consequently, when woking exclusively in CGS EMU, equation is often witten as p B = This equation appeas not to balance dimensionally. Howeve, the equation is not witten accoding to the conventions of quantity calculus, and the symbols do not stand fo physical quantities. Rathe, they stand fo thei numeical values in a paticula system of units. Thus is the distance in cm, B is the field in gauss, and p is the magnetic moment in dyne cm pe gauss. Howeve, because of the deceptive appeaance of the equation, a common pactice, fo example, in calculating the magnetic moment of a planet is to measue its suface equatoial field, multiply it by the cube of the planet s adius, and then quote the magnetic moment in G cm. While the numeical esult is coect fo the magnetic moment in CGS EMU, the units quoted ae not. While some may conside objections to incoect units to be mee pedanty (and who would pesumably theefoe see nothing wong with quoting a length in gams, as long as the actual numbe is coect), the situation becomes moe difficult when a wite, wishing to pay lip sevice to SI, attempts to use equation using SI units, by multiplying the suface equatoial field in T by the cube of the planet s adius, and then giving the magnetic moment in T m, a clealy disastous ecipe! Of couse, some may use equation as a definition of magnetic moment. If that is so, then the quantity so defined is clealy not the same quantity, physically, conceptually, dimensionally o numeically, as the quantity defined as magnetic moment in Section Possible Altenative Definitions of Magnetic Moment Although the standad SI definition of magnetic moment is descibed in Section 17.2, and thee is little eason fo anyone who wishes to be undestood by othes to use any othe, the pevious paagaph suggested that thee might be moe than one choice as to how one wishes to define magnetic moment. Do we use equation o equation as the definition? (They ae clealy diffeent concepts.) Additional degees of feedom as to how one might choose to define magnetic moment depend on whethe we elect to use

4 4 magnetic field H o magnetic field B in the definition, o whethe the pemeability is o is not to include the facto 4π in its definition that is, whethe we elect to use a ationalized o unationalized definition of pemeability. If one chooses to define the magnetic moment as the maximum toque expeienced in unit extenal magnetic field, thee is still a choice as to whethe by magnetic field we choose H o B. Thus we could define magnetic moment by eithe of the following two equations: τ = p 1 H o τ = p B Altenatively, we could choose to define the magnetic moment is tems of the field on the equato. In that case we have a choice of fou. We can choose to use B o H fo the magnetic field, and we can choose to exclude o include 4π in the denominato: B = p, H = p4, B = p5 4π, H = p6. 4π These six possible definitions of magnetic moment ae clealy diffeent quantities, and one may well wonde why to list them all. The eason is that all of them ae to be found in cuent scientific liteatue. To give some hint as to the unnecessay complications intoduced when authos depat fom the simple SI definition, I list in Table XVII.1 the dimensions of each vesion of magnetic moment, the CGS EM unit, the SI unit, and the convesion facto between CGS and SI. The convesion factos cannot be obtained simply by efeing to the dimensions, because this does not take into account the inclusion o exclusion of 4π in the pemeability. The coect factos can be obtained fom the units, fo example by noting that 1 Oe = 10 /(4π) A m 1 and 1 G = 10 4 T.

5 5 TABLE XVII.1 DIMENSIONS, CGS AND SI UNITS, AND CONVERSION FACTORS FOR MAGNETIC MOMENTS Dimensions 1 CGS EMU = Convesion facto SI unit p 1 ML T 1 Q 1 1 dyn cm Oe 1 = 4π N m (A/m) 1 p 2 L 2 T 1 Q 1 dyn cm G 1 = 10 N m T 1 p ML T 1 Q 1 1 G cm = T m p 4 L 2 T 1 Q 1 Oe cm = 10 /4π A m 2 p 5 ML T 1 Q 1 1 G cm = T m p 6 L 2 T 1 Q 1 Oe cm = 10 /4π A m Thiteen Questions We have seen that the SI definition of magnetic moment is unequivocally defined as the maximum toque expeienced in unit extenal field. Nevetheless some authos pefe to think of magnetic moment as the poduct of the equatoial field and the cube of the distance. Thus thee ae two conceptually diffeent concepts of magnetic moment, and, when to these ae added mino details as to whethe the magnetic field is B o H, and whethe o not the pemeability should include the facto 4π, six possible definitions of magnetic moment, descibed in Section 17.6, all of which ae to be found in cuent liteatue, aise. Regadless, howeve, how one chooses to define magnetic moment, whethe the SI definition o some othe unconventional definition, it should be easily possible to answe both of the following questions: A. Given the magnitude of the equatoial field on the equato of a magnet, what is the maximum toque that that magnet would expeience if it wee placed in an extenal field? B. Given the maximum toque that a magnet expeiences when placed in an extenal field, what is the magnitude of the equatoial field poduced by the magnet? It must suely be conceded that a failue to be able to answe such basic questions indicates a failue to undestand what is meant by magnetic moment.

6 6 I theefoe now ask a seies of thiteen questions. The fist six ae questions of type A, in which I use the six possible definitions of magnetic moment. The next six ae simila questions of type B. And the last is an absudly simple question, which anyone who believes he undestands the meaning of magnetic moment should easily be able to answe. 1. The magnitude of the field in the equatoial plane of a magnet at a distance of 1 cm is 1 Oe. field of 1 Oe, and what is its magnetic moment? Note that, in this question and the following seven thee must be a unique answe fo the toque. The answe you give fo the magnetic moment, howeve, will depend on how you choose to define magnetic moment, and on whethe you choose to give the answe in SI units o CGS EMU. 2. The magnitude of the field in the equatoial plane of a magnet at a distance of 1 cm is 1 Oe. field of 1 G, and what is its magnetic moment?. The magnitude of the field in the equatoial plane of a magnet at a distance of 1 cm is 1 G. field of 1 Oe, and what is its magnetic moment? 4. The magnitude of the field in the equatoial plane of a magnet at a distance of 1 cm is 1 G. field of 1 G, and what is its magnetic moment? 5. The magnitude of the field in the equatoial plane of a magnet at a distance of 1 m is 1 A m 1. field of 1 A m 1, and what is its magnetic moment?

7 7 6. The magnitude of the field in the equatoial plane of a magnet at a distance of 1 m is 1 A m 1. field of 1 T, and what is its magnetic moment? 7. The magnitude of the field in the equatoial plane of a magnet at a distance of 1 m is 1 T. field of 1 A m 1, and what is its magnetic moment? 8. The magnitude of the field in the equatoial plane of a magnet at a distance of 1 m is 1 T. field of 1 T, and what is its magnetic moment? 9. A magnet expeiences a maximum toque of 1 dyn cm if placed in a field of 1 Oe. What is the magnitude of the field in the equatoial plane at a distance of 1 cm, and what is the magnetic moment? Note that, in this question and the following thee thee must be a unique answe fo B and a unique answe fo H, though each can be expessed in SI o in CGS EMU, while the answe fo the magnetic moment depends on which definition you adopt. 10. A magnet expeiences a maximum toque of 1 dyn cm if placed in a field of 1 G. What is the magnitude of the field in the equatoial plane at a distance of 1 cm, and what is the magnetic moment? 11. A magnet expeiences a maximum toque of 1 N m if placed in a field of 1 A m 1. What is the magnitude of the field in the equatoial plane at a distance of 1 m, and what is the magnetic moment? 12. A magnet expeiences a maximum toque of 1 N m if placed in a field of 1 T. What is the magnitude of the field in the equatoial plane at a distance of 1 m, and what is the magnetic moment? I ll pose Question Numbe 1 a little late. In the meantime the answes to the fist fou questions ae given in Table XVII.2, and the answes to Questions 5 12 ae given in

8 8 Tables XVII. and 4. The shee complexity of these answes to absudly simple questions is a consequence of diffeent usages by vaious authos of the meaning of magnetic moment and of depatue fom standad SI usage. TABLE XVII.2 ANSWERS TO QUESTIONS 1 4 IN CGS EMU AND SI UNITS The answes to the fist fou questions ae identical τ = 1 dyn cm = 10 7 N m p 1 = 1 dyn cm Oe 1 = 4π 10 7 N m (A/m) 1 p 2 = 1 dyn cm G 1 = 10 N m (T) 1 p = 1 G cm = T m p 4 = 1 Oe cm = 10 /(4π) A m 2 p 5 = 4π G cm = 4π T m p 6 = 4π Oe cm = 10 A m 2

9 9 TABLE XVII. ANSWERS TO QUESTIONS 5 8 IN CGS EMU AND SI UNITS τ = 2 ( 4π ) 4π π dyn cm = (4π) π 4π 10 7 N m p l = 4π 10 4π dyn cm Oe 1 = (4π) (4π) π 4π N m (A/m) 1 p 2 = 4π 10 4π dyn cm G 1 = 4π 4π N m T 1 p = 4π 10 4π G cm = 4π π T m p 4 = 4π 10 4π Oe cm = /(4π) 10 7 /(4π) A m 2 p 5 = (4π) 2 10 (4π) π π G cm = (4π) (4π) π 4π T m p 6 = (4π) 2 10 (4π) π π Oe cm = 4π 4π A m 2

10 10 TABLE XVII.4 ANSWERS TO QUESTIONS 9 12 IN CGS EMU AND SI UNITS B = /(4π) 10 G = /(4π) 10 7 T H = /(4π) 10 Oe = 10 /(4π) 10 /(4π) 10 7 /(4π) 2 1/(4π) A m 1 p l = /(4π) 10 dyn cm Oe 1 = 4π π π 10 7 N m (A/m) 1 p 2 = /(4π) 10 dyn cm G 1 = /(4π) 1 N m T 1 p = /(4π) 10 G cm = /(4π) 10 1 T m p 4 = /(4π) 10 Oe cm = 10 /(4π) 10 /(4π) 10/(4π) /(4π) A m 2 p 5 = 4π 4π π 10 G cm = 4π π π 10 1 T m p 6 = 4π 4π π 10 Oe cm = /(4π) 10 6 A m 2

11 11 The thiteenth and last of these questions is as follows: Assume that Eath is a sphee of adius m = cm, and that the suface field at the magnetic equato is 5 1 B = 10 T = 0. G, o H = 75/ π A m = 0. Oe, what is the magnetic moment of Eath? It is had to imagine a moe staightfowad question, yet it would be had to find two people who would give the same answe. The SI answe (which, to me, is the only answe) is µ p 4π B 4π ( ) B =, p = = = N m T π µ 0 4π 10 This esult coectly pedicts that, if Eath wee placed in an extenal field of 1 T, it would expeience a maximum toque of N m, and this is the nomal meaning of what is meant by magnetic moment. A calculation in GCS might poceed thus: p 8 B = p = B = ( ) 0. = , G cm. Is this the same esult as was obtained fom the SI calculation? We can use the convesions 1 G = 10 4 T and 1 cm = 10 6 m, and we obtain p = T m. We aive at a numbe that not only diffes fom the SI calculation by 10 7, but is expessed in quite diffeent, dimensionally dissimila, units. Pehaps the CGS calculation should be p 8 H = p = H = ( ) 0. = , 25 Now 1 Oe = 1000/(4π) A m 1 and 1 cm = 10 6 m, and we obtain p = A m 2 Oecm. This time we aive at SI units that ae dimensionally simila to N m T 1, and which ae pefectly coect SI units, but the magnetic moment is smalle than coectly pedicted by the SI calculation by a facto of Yet again, we might do what appeas to be fequently done by planetay scientists, and we can multiply the suface field in T by the cube of the adius in m to obtain

12 12 p = T m. This aives at the same esult as one of the CGS calculations, but, whateve it is, it is not the magnetic moment in the sense of the geatest toque in a unit field. The quantity so obtained appeas to be nothing moe that the poduct of the suface equatoial field and the cube of the adius, and as such would appea to be a puposeless and meaningless calculation. It would be a good deal moe meaningful meely to multiply the suface value of H by. This in fact would give (coectly) the dipole moment divided by the volume of Eath, and hence it would be the aveage magnetization of Eath a vey meaningful quantity, which would be useful in compaing the magnetic popeties of Eath with those of the othe planets Additional Remaks The units eg G 1 o J T 1 ae fequently encounteed fo magnetic moment. These may be dimensionally coect, although egs and joules (units of enegy) ae not quite the same things as dyn cm o N m as units of toque. It could be agued that magnetic moment could be defined fom the expession p.b fo the potential enegy of a magnet in a magnetic field. But the coect expession is actually constant p.b, the constant being zeo only if you specify that the enegy is taken to be zeo when the magnetic moment and field vectos ae pependicula to each othe. This seems meely to add yet futhe complications to what should be, but unfotunately is not, a concept of the utmost simplicity. Nevetheless the use of egs o joules athe than dyn cm o N m is not uncommon, and nuclea and paticle physicists commonly convet joules to MeV. Magnetic moments of atomic nuclei ae commonly quoted in nuclea magnetons, whee a nuclea magneton is eh /( 2mp ) and has the value MeV T 1. While one is neve likely to want to expess the magnetic moment of the planet Uanus in nuclea magetons, it is sobeing to attempt to do so, given that the magnetic moment of Uanus is quoted as 0.42 Oe km 1. While on the subject of Uanus, I have seen it stated that the magnetic quadupole of Uanus is o the same ode of magnitude as its magnetic dipole moment though, since these ae dimensionally dissimila quantities, such a statement conveys no meaning. Anothe execise to illustate the points I have been tying to make is as follows. Fom fou published papes I find the following. The magnetic moment of Mecuy is A m 2 in one pape, and 00 nt R M in anothe. The magnetic moment of Uanus is Oe km in one pape, and 0.2 G R U in anothe. The adii of Mecuy and Uanus ae, espectively, m and m. Calculate the atio of the magnetic moment of Uanus to that of Mecuy. If you ae by now completely confused, you ae not alone.

13 Conclusion Reades will by now pobably be bewildeed at the complexities descibed in this chapte. Afte all, thee could scacely be a simple notion than that of the toque expeienced by a magnet in a magnetic field, and thee would seem to be no need fo all of these complicated vaiations. You ae ight thee is no such need. All that need be known is summaized in Sections 17.2 and 17.. The difficulty aises because authos of scientific papes ae using almost all possible vaiations of what they think is meant by magnetic moment, and this has led to a thooughly chaotic situation. All I can do is to hope that eades of these notes will be encouaged to use only the standad SI definition and units fo magnetic moment, and to be awae of the enomous complications aising when they depat fom these.